Remark Besides their role in generic regularity of area-minimizing hypersurfaces in eight-dimensional manifolds, the surfaces \(S_\pm \)  are important objects in their own right, cf. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. In our previous paper [7]}, we proved the parabolic analogue of Theorem REF  (for compact/asymptotically-conical self-shrinkers) by constructing and classifying ancient one-sided flows analogous to the surfaces \(S_{\pm }\) .

